The value of \(11.\overline{4}\) + \(22.5\overline{67}\) – \(33.5\overline{9}\) is:
Calculating Expressions with Repeating Decimals
To find the value of an expression involving repeating decimals, it is often easiest to convert each repeating decimal into a fraction. Once converted, standard fraction arithmetic can be performed.
Converting Repeating Decimals to Fractions
Let's convert each term in the expression \(11.\overline{4} + 22.5\overline{67} - 33.5\overline{9}\) into a fraction.
Term 1: \(11.\overline{4}\)
Let \(x = 11.\overline{4}\). The repeating part is '4', which has 1 digit. Multiply by \(10^1 = 10\).
- \(x = 11.444...\)
- \(10x = 114.444...\)
- Subtracting the first equation from the second:
- \(10x - x = 114.444... - 11.444...\)
- \(9x = 103\)
- \(x = \frac{103}{9}\)
Term 2: \(22.5\overline{67}\)
Let \(y = 22.5\overline{67}\). This number has a non-repeating part '5' (1 digit) and a repeating part '67' (2 digits). First, move the decimal past the non-repeating part by multiplying by \(10^1 = 10\).
- \(y = 22.5676767...\)
- \(10y = 225.676767...\)
Now, the number after the decimal in \(10y\) is \(25.676767...\). The repeating part is '67' (2 digits). Multiply \(10y\) by \(10^2 = 100\).
- \(100 \times 10y = 1000y = 22567.676767...\)
- Subtract \(10y\) from \(1000y\):
- \(1000y - 10y = 22567.676767... - 225.676767...\)
- \(990y = 22342\)
- \(y = \frac{22342}{990}\)
- Simplifying the fraction by dividing both numerator and denominator by 2:
- \(y = \frac{11171}{495}\)
Term 3: \(33.5\overline{9}\)
Let \(z = 33.5\overline{9}\). The repeating part is '9' (1 digit). Multiply by \(10^1 = 10\).
- \(z = 33.5999...\)
- \(10z = 335.999...\)
- Subtracting the first equation from the second:
- \(10z - z = 335.999... - 33.5999...\)
- \(9z = 302.4\)
- \(z = \frac{302.4}{9} = \frac{3024}{90}\)
- Simplifying the fraction:
- \(\frac{3024}{90} = \frac{3024 \div 18}{90 \div 18} = \frac{168}{5}\)
Alternatively, note that \(0.\overline{9} = 1\). So, \(33.5\overline{9} = 33.5 + 0.0\overline{9} = 33.5 + 0.1 = 33.6\). As a fraction, \(33.6 = \frac{336}{10} = \frac{168}{5}\).
Performing the Calculation with Fractions
The expression is now \(\frac{103}{9} + \frac{11171}{495} - \frac{168}{5}\).
To add and subtract these fractions, we need a common denominator. The denominators are 9, 495, and 5.
- Prime factorization of denominators: \(9 = 3^2\), \(495 = 5 \times 99 = 5 \times 3^2 \times 11\), \(5 = 5^1\).
- The least common multiple (LCM) is \(3^2 \times 5 \times 11 = 9 \times 5 \times 11 = 495\).
Convert each fraction to have the denominator 495:
- \(\frac{103}{9} = \frac{103 \times (495 \div 9)}{9 \times (495 \div 9)} = \frac{103 \times 55}{9 \times 55} = \frac{5665}{495}\)
- \(\frac{11171}{495}\) (already has the common denominator)
- \(\frac{168}{5} = \frac{168 \times (495 \div 5)}{5 \times (495 \div 5)} = \frac{168 \times 99}{5 \times 99} = \frac{16632}{495}\)
Now substitute these back into the expression:
\(\frac{5665}{495} + \frac{11171}{495} - \frac{16632}{495} = \frac{5665 + 11171 - 16632}{495}\)
Calculate the numerator:
- \(5665 + 11171 = 16836\)
- \(16836 - 16632 = 204\)
The resulting fraction is \(\frac{204}{495}\).
Simplifying the Resulting Fraction
The fraction \(\frac{204}{495}\) can be simplified. Both numerator and denominator are divisible by 3 (sum of digits of 204 is 6, sum of digits of 495 is 18).
\(\frac{204 \div 3}{495 \div 3} = \frac{68}{165}\)
Let's check for other common factors. Factors of 68 are 1, 2, 4, 17, 34, 68. Factors of 165 are 1, 3, 5, 11, 15, 33, 55, 165. The only common factor is 1. The fraction is simplified.
Converting the Result Back to a Decimal
Now, we convert the simplified fraction \(\frac{68}{165}\) back to a decimal by performing division \(68 \div 165\).
| Division Step | Calculation | Result / Remainder |
|---|---|---|
| 68 ÷ 165 | 68 < 165 | 0. (Remainder 68) |
| 680 ÷ 165 | 165 × 4 = 660 | 4 (Remainder 680 - 660 = 20) |
| 200 ÷ 165 | 165 × 1 = 165 | 1 (Remainder 200 - 165 = 35) |
| 350 ÷ 165 | 165 × 2 = 330 | 2 (Remainder 350 - 330 = 20) |
| 200 ÷ 165 | 165 × 1 = 165 | 1 (Remainder 200 - 165 = 35) |
The remainders are 20, 35, 20, 35, ... . The sequence of digits in the quotient after the first digit '4' is 12, 12, ... .
Thus, the decimal representation of \(\frac{68}{165}\) is \(0.4121212... = 0.4\overline{12}\).
Conclusion
The value of the expression \(11.\overline{4} + 22.5\overline{67} - 33.5\overline{9}\) is \(0.4\overline{12}\).
Revision Table: Working with Repeating Decimals
| Concept | Description | Example |
|---|---|---|
| Pure Repeating Decimal | A decimal where all digits after the decimal point repeat. | \(0.\overline{3} = 0.333...\) |
| Mixed Repeating Decimal | A decimal with a non-repeating part followed by a repeating part. | \(0.1\overline{27} = 0.1272727...\) |
| Conversion (Pure) | \(0.\overline{a_1a_2...a_n} = \frac{a_1a_2...a_n}{\underbrace{99...9}_{n \text{ times}}}\) | \(0.\overline{6} = \frac{6}{9} = \frac{2}{3}\) |
| Conversion (Mixed) | \(0.b_1...b_m\overline{a_1...a_n} = \frac{b_1...b_m a_1...a_n - b_1...b_m}{\underbrace{99...9}_{n \text{ times}}\underbrace{00...0}_{m \text{ times}}}\) | \(0.1\overline{2} = \frac{12-1}{90} = \frac{11}{90}\) |
Additional Information: Properties of Repeating Decimals
Repeating decimals are rational numbers. This means they can always be expressed as a fraction \(\frac{p}{q}\), where p and q are integers and q is not zero. The process of converting repeating decimals to fractions demonstrates this property.
Understanding the structure of repeating decimals helps in performing arithmetic operations. While direct addition/subtraction can be complicated, conversion to fractions provides a standard method.
A decimal terminates if and only if the prime factors of the denominator of its simplified fraction form are only 2 and 5. If the denominator has other prime factors, the decimal representation will be repeating.
In our case, the denominator of the simplified fraction \(\frac{68}{165}\) is 165. The prime factorization of 165 is \(3 \times 5 \times 11\). Since there are prime factors other than 2 and 5 (namely 3 and 11), the decimal representation is indeed repeating.
Important Questions from Decimals
The value of \(\frac{1}{4} + \frac{{[{{(20.35)}^2} - {{(8.35)}^2}] \times 0.0175}}{{{{(1.05)}^2} + (1.05)(27.65)}}\) is:
The value of \(0.4\overline 6 + 0.7\overline {23} - 0.3\overline 9 \times 0.\overline 7 \) is:
The value of \(\frac{48.3\times[(4.95)^2+4.95\times13.25]}{[(12.55)^2-(5.65)^2]\times19.8} \) is:
Find the value of (1.6) 3 - (0.9) 3 - (0.7) 3.
What is the value of x, if \(5\left( {1 - \frac{x}{5}} \right) - (5 - x) - \frac{1}{{200}}{\rm{of (20 - x) = 0}}{\rm{.08}}\) ?
